- #1

JamesGoh

- 143

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For a 3x3 orthogonal matrix with determinant= -1 (which means rotation followed by simple reflection), is the axis of rotation the same as the plane of reflection ?

My reasoning is follows (see attachment)

Say you have two vectors with the same angle size (which i call A), same x-values, but one of the z-values (height in this case), is the negative of the other (n.b. y-value is zero in both vectors )

Vector 1 rotates around the x-axis by 2A to get to the same spot as vector 2. Because the angle size is the same and because the z-component of vector 2 is the negative of the z-component of vector 1, we get a reflection ?

Since the "reflection" happens about the x-axis, this is why the plane of reflection is the same as the axis of rotation in the case of a 3x3 orthogonal matrix having determinant 1 ?

My reasoning is follows (see attachment)

Say you have two vectors with the same angle size (which i call A), same x-values, but one of the z-values (height in this case), is the negative of the other (n.b. y-value is zero in both vectors )

Vector 1 rotates around the x-axis by 2A to get to the same spot as vector 2. Because the angle size is the same and because the z-component of vector 2 is the negative of the z-component of vector 1, we get a reflection ?

Since the "reflection" happens about the x-axis, this is why the plane of reflection is the same as the axis of rotation in the case of a 3x3 orthogonal matrix having determinant 1 ?